L(s) = 1 | + (1.11 + 2.19i)3-s − 2.23·5-s + (2.71 + 2.71i)7-s + (−1.79 + 2.47i)9-s + (−1.87 − 2.57i)11-s + (0.951 + 6.00i)13-s + (−2.49 − 4.90i)15-s + (−2.48 − 1.26i)17-s + (−0.812 − 2.49i)19-s + (−2.91 + 8.97i)21-s + (−0.902 + 5.70i)23-s + 5.00·25-s + (−0.147 − 0.0232i)27-s + (−3.50 − 1.13i)29-s + (6.62 − 2.15i)31-s + ⋯ |
L(s) = 1 | + (0.645 + 1.26i)3-s − 0.999·5-s + (1.02 + 1.02i)7-s + (−0.599 + 0.825i)9-s + (−0.564 − 0.777i)11-s + (0.263 + 1.66i)13-s + (−0.645 − 1.26i)15-s + (−0.603 − 0.307i)17-s + (−0.186 − 0.573i)19-s + (−0.636 + 1.95i)21-s + (−0.188 + 1.18i)23-s + 1.00·25-s + (−0.0283 − 0.00448i)27-s + (−0.650 − 0.211i)29-s + (1.19 − 0.386i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.772591 + 1.21740i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.772591 + 1.21740i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
good | 3 | \( 1 + (-1.11 - 2.19i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-2.71 - 2.71i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.87 + 2.57i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.951 - 6.00i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.48 + 1.26i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.812 + 2.49i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.902 - 5.70i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (3.50 + 1.13i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.62 + 2.15i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (8.82 - 1.39i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-1.43 - 1.04i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-3.91 + 3.91i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.95 + 4.05i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-5.97 + 3.04i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-2.77 - 2.01i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.82 + 2.05i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.49 - 4.90i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-14.7 - 4.78i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.39 - 1.01i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-1.37 + 4.22i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-8.26 - 4.20i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (6.66 + 9.16i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (3.29 + 6.46i)T + (-57.0 + 78.4i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.42800839118726033410832019013, −10.84819101305882705160804209676, −9.519917848867142652811199656635, −8.718569663109807655621416430198, −8.354420300014023752608652776480, −7.04429941969110652982695189474, −5.43878344339470589438401765469, −4.52385257787605672085679409403, −3.70704767782284509641134963106, −2.36845049353496537851339629053,
0.934608939440140891723968181892, 2.44253989170873814949486212526, 3.86729725540675477864064478486, 4.99451092828624852126285105822, 6.63831214557327055350026928724, 7.60511901975763772561477836683, 7.919654015685348681244511021724, 8.608480002006340850925575318996, 10.47274971713873203213808634692, 10.81704050656930338962045482861